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removed most of the difficulties which Legendre had grappled
with for forty years. The true theory of these important inte-
grals rushed forth almost of itself after this obstruction had
been removed - like a log-jam going down the river after the
M"g log has been snaked out.
When Legendre grasped what Abel and Jacobi had done he
encouraged them most cordially, although he realized that their
simpler approach (that of inversion) nullified what was to have
been his own masterpiece of forty years' labour. For Abel, alas,
Legendre's praise came too late, but for Jacobi'it was an
inspiration to surpass himself. In one of the finest correspon-
dences in the whole of scientific literature the young man in his
early twenties and the veteran in his late seventies strive to
outdo one another in sincere praise and gratitude. The only
jarring note is Legendre's outspoken disparagement of Gauss,
whom Jacobi vigorously defends. But as Gauss never con-
descended to publish his researches - he had planned a major
work on elliptic functions when Abel and Jacobi anticipated
him in publication - Legendre can hardly be blamed for holding
a totally mistaken opinion. For lack of space we must omit
extracts from this beautiful correspondence (the letters are
given in full in vol. 1 of Jacobi's Werke - in French).
The joint creation with Abel of the theory of elliptic functions
was only a small if highly important part of Jacobi's huge out-
put. Only to enumerate all the fields he enriched in his brief
working life of less than a quarter of a century would take more
space than can be devoted to one man hi an account like the
present, so we shall merely mention a few of the other great
things he did.
Jacobi was the first to apply elliptic functions to the theory
of numbers. This was to become a favourite diversion with some
of the greatest mathematicians who followed Jacobi. It is a
curiously recondite subject, where arabesques of ingenious
algebra unexpectedly reveal hitherto unsuspected relations
between the common whole numbers. It was by this means that
Jacobi proved the famous assertion of Fermat that every
integer 192,3, ... is a sum of four integer squares (zero being
counted as an integer) and, moreover, bis beautiful analysis